Time-optimal control strategies in SIR epidemic models

Abstract

We investigate the time-optimal control problem in SIR (Susceptible-Infected-Recovered) epidemic models, focusing on different control policies: vaccination, isolation, culling, and reduction of transmission. Applying the Pontryagin's Minimum Principle (PMP) to the unconstrained control problems (i.e. without costs of control or resource limitations), we prove that, for all the policies investigated, only bang-bang controls with at most one switch are admitted. When a switch occurs, the optimal strategy is to delay the control action some amount of time and then apply the control at the maximum rate for the remainder of the outbreak. This result is in contrast with previous findings on the unconstrained problems of minimizing the total infectious burden over an outbreak, where the optimal strategy is to use the maximal control for the entire epidemic. Then, the critical consequence of our results is that, in a wide range of epidemiological circumstances, it may be impossible to minimize the total infectious burden while minimizing the epidemic duration, and vice versa. Moreover, numerical simulations highlighted additional unexpected results, showing that the optimal control can be delayed also when the control reproduction number is lower than one and that the switching time from no control to maximum control can even occur after the peak of infection has been reached. Our results are especially important for livestock diseases where the minimization of outbreaks duration is a priority due to sanitary restrictions imposed to farms during ongoing epidemics, such as animal movements and export bans.

Modelling lockdown measures in epidemic outbreaks using selective socio-economic containment with uncertainty

“…Solving the above optimization problem, however, is generally quite complicated and computationally demanding when there are uncertainties as it involves solving simultaneously the forward problem (4)- (7) and the backward problem derived from the optimality conditions [4]. Furthermore, the assumption that the policy maker follows an optimal strategy over a long time horizon seems rather unrealistic in the case of a rapidly spreading disease such as the COVID-19 epidemic.…”

Optimal impulse control of a SIR epidemic

“…A brief review of applications of control theory to infectious disease models appears in the work of Clancy. 2 Control can mean vaccination of succeptibles [3][4][5] or isolation/treatment of infectives. 4,6,7 In the cited papers, the control was gradual, meaning that it affects the dynamics of the system; the trajectories remain continuous.…”

“…2 Control can mean vaccination of succeptibles [3][4][5] or isolation/treatment of infectives. 4,6,7 In the cited papers, the control was gradual, meaning that it affects the dynamics of the system; the trajectories remain continuous. The optimal control strategy was obtained using the dynamic programming 5,6 and the maximum principle.…”

“…The optimal control strategy was obtained using the dynamic programming 5,6 and the maximum principle. 3,4,7 Impulse control means that at any time it is possible, for a certain price, to reduce instantly the number of susceptibles (case of vaccination) or infectives (case of isolation). Such models, the case of isolation, were investigated in other works.…”

Dynamics of a metapopulation epidemic model with localized culling

“…Specifically, by reducing the host population abundance, culling can reduce the density-dependent constrains on host birth rate, thereby producing a flush in new susceptible individuals in the population [29,46]. These new susceptibles represent a reservoir for new infections, which nullifies the expected benefits of disease control campaigns or, in some cases, even increases the disease burden in the population [5,7,15] or the duration of the epidemic [9,10].…”